2015 AMC 10A Problems/Problem 2
A box contains a collection of triangular and square tiles. There are tiles in the box, containing edges total. How many square tiles are there in the box?
Let be the amount of triangular tiles and be the amount of square tiles.
Triangles have edges and squares have edges, so we have a system of equations.
We have tiles total, so .
We have edges total, so .
Multiplying the first equation by on both sides gives .
Second equation minus the first equation gives , so the answer is .
If all of the tiles were triangles, there would be edges. This is not enough, so there needs to be some squares. Trading a triangle for a square results in one additional edge each time, so we must trade out triangles for squares. Answer:
Let be the number of square tiles. A square has edges, so the total number of edges from the square tiles is . There are total tiles, which means that there are triangle tiles. A triangle has edges, so the total number of edges from the triangle tiles is . Together, the total number of edges is . Solving our equation, we get that which means that our answer is .
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